Cusps in $\eta\prime\rightarrow \eta \pi \pi$ decays

Abstract

The discovery of the cusp effect in the decay K + → π + π 0 π 0 has spurred the search for other decay channels, where this phenomenon, which is generated by strong final-state interactions, should also occur. A very promising candidate is η ′ → ηπ 0 π 0 . The cusp effect offers an excellent opportunity to experimentally extract ππ S-Wave scattering lengths. We adapt and generalize the non-relativistic effective field theory framework developed for K → 3π decays to η ′ → ηππ. The cusp effect is predicted to have an effect of more than 8% on the decay spectrum below the π + π − threshold [1]. We also show that with our current theoretical information about η ′ → ηππ decays, it is not possible to extract πη threshold parameters.

Cusp effects in meson decays

“…From what we have learnt so far, such a cusp above threshold would have to be a two-loop effect and therefore is expected to be small. However, things turn our to be even worse: with the use of the threshold theorem again, applied now to the threshold s 1 = (M η + M π ) 2 , one can show that the interference of genuine two-loop graphs with the tree-level amplitude (both real) is always exactly cancelled by the product of two corresponding one-loop graphs (both imaginary in our formalism), such that no square-root behavior survives in the squared amplitude [67]. This cancellation is shown 01008-p.10 schematically in Fig.…”

Dispersion relations for $$\eta '\rightarrow \eta \pi \pi $$

“…A fifth subtraction constant would be introduced if we assumed a different high-energy behavior of δ 1 1 : if a resonance with exotic quantum numbers J PC = 1 −+ coupling to πη exists (the search for which seems inconclusive so far [38,39]) and we assume the P-wave phase to approach π instead of 0 asymptotically, the P-wave would allow for a nonvanishing (constant) subtraction polynomial in Eq. (17), which cannot be removed by the transformation (22).…”

“…The hope of extracting scattering parameters from a two-loop cusp is shattered likewise: there is a rather subtle cancellation of this effect at threshold (see Refs. [22,23] for an elaborate discussion).…”

Meson decay studies from MAMI A2

“…Several theoretical approaches have been used to model this rescattering effect and allowed to extract the S-wave ππ scattering length combination, a 0 − a 2 . Using a nonrelativistic effective field theory (NREFT) framework, this effect has been predicted be of 1% magnitude for η → π 0 π 0 π 0 [11] and of 6% magnitude for η ′ → ηπ 0 π 0 [12].…”